Animation changes after applying mirror modifier
The object has an origin which is used as center for the rotation but your object is made by doubling mirror on object so the initial object's origin is being used as the generated object's origin by the animation. 

To fix this you must separate the mesh obtained after applying mirrors into 4 meshes, into edit mode select the faces of one propeller, press "P" and hit "selection", you must do this three times so that you have your four propellers.

Once this is done you still have one thing to do, it is to set your 4 objects origins: use "Shit-S" to access the alignment tools and use it in edit mode to set your cursor precisely where you want your rotation axis to be. 

Then back to object mode "object>tranform>origin to 3d cursor", you could also use origin to geometry (in this case blender will interpolate between all the vertices of your object and set its origin there) and note that geometry to origin is the same as "apply location